2022 RI J1 CT Q8
The earth can be modelled as a sphere, centered at the origin $O$ with radius $r$.
A point on the sphere with longitude $\varphi $ and latitude $\theta $, both measured in degrees, has position vector $\overrightarrow{OP}=\left( \begin{matrix}
r\sin \theta \cos \varphi \\
r\sin \theta \sin \varphi \\
r\cos \theta \\
\end{matrix} \right)$.
The following points on the sphere have the corresponding longitudes and latitudes, measured in degrees:
(i)
Show that the cartesian equation of the plane containing $O$, $F$ and $R$ is given by $\sqrt{3}y=x$.
[3]
A spherical angle is the angle between two arcs on a sphere, and is measured by the acute angle between the planes containing the arcs and the centre of the sphere.
(ii)
Find the spherical angle between the arcs $FR$ and $IC$.
[4]
An aircraft travels from $F$ to $C$ along the arc $FC$. Let $\alpha $ be the angle between the vectors $\overrightarrow{OF}$ and $\overrightarrow{OC}$.
(iii)
Show that $\overrightarrow{OF}\cdot \overrightarrow{OC}={{r}^{2}}\cos \alpha $.
[1]
(iv)
Hence find the length of the arc $FC$ to the nearest km, given that the radius of earth can be taken to be $6378$km.
[3]
(v)
Give a reason why the actual distance travelled by the aircraft is usually greater than the arc length $FC$ calculated in part (iv).
[1]
Suggested Video and Handwritten Solutions
Share with your friends!
- No charge
- Unsubscribe anytime
Continue reading
Maximizing June Holidays: Is Joining a Crash Course Necessary?
As June holidays approach, students find themselves at a crucial juncture in their academic journey. With the semester winding down and exams looming on the horizon, the pressure to excel
5 Exam Preparation Tips: How We Help Students Excel in Exams
In the ever-evolving landscape of education, where academic success is often measured by performance in exams, the need for effective test preparation has never been more critical. Students face a
List MF 27
What is the MF 27? The MF 27, set to replace the MF 26 from 2025, is a comprehensive formula sheet developed by the Ministry of Education Singapore in collaboration